Note that the emergence of bistable region is a new physical phenomenon arising from multilayer reinforcement. Here, we further explore the detailed conditions of phase transition for bistability. In Fig. REF , we present phase diagram under different combinations of multilayer reinforcement and intralayer transmissibility (Fig. REF (a)) or combinations of multilayer reinforcement and interlayer transmissibility. (Fig. REF (b)). In all subfigures, The phase plane is divided into three regions by two separatrixes (\(L_1\)  and \(L_2\) ): outbreak, bistable, extinction. The outbreak state means that the prevalence is positive as long as \(\rho _0>0\) , and the extinction state represents that the prevalence vanishes no matter what \(\rho _0\)  is. The bistable state means that norms outbreak for large \(\rho _0\)  while go extinct for small \(\rho _0\) . Thus, the two separatrixes \(L_1\)  and \(L_2\)  are critical parameters under very small \(\rho _0\)  and \(\rho _0=1\) , which can be numerically solved by Eq. (REF ) and Eqs. (REF ), respectively. We find that the solution of Eq. (REF ) is uncorrelated with multilayer reinforcement (\(\gamma \) ) which is directly reflected by the parallel relationship between \(L_1\)  and y-axis. Meanwhile, \(L_2\)  is determined by the joint effects of intralayer transmissibility, interlayer transmissibility and multilayer reinforcement. To sum up, region 1 and region 2 highlight two opportunities for outbreak, which are: (i) to own large transmissibilities and (ii) to own large initial density of adopters with strong multilayer reinforcement. These conclusions can be directly applied to many scenarios such as promoting marketing, designing interventions for rumor spreading and establishing new social norms [1]}, [2]}, [3]}.
<FIGURE>